Non-reciprocal circular dichroism of ferro-rotational phonons in MnTiO$_{3}$

Abstract

X-ray circular dichroism (XCD), defined as the difference in absorption or scattering intensity between X-rays of opposite polarizations, arises from the breaking of spatial inversion or time-reversal symmetry and is thus sensitive to chirality, magnetism, and their interplay. Non-reciprocal XCD, in which the dichroic response changes upon reversing the propagation direction of the probe, is generally forbidden in systems with both symmetries. Using resonant inelastic X-ray scattering, we identify circularly polarized phonons in ferro-rotational MnTiO$_3$, which we term ferro-rotational phonons. Their excitations provide a direct demonstration of non-reciprocal XCD in a system that globally preserves inversion and time-reversal symmetries. We propose that a condensate of these phonons, manifested as standing waves, underlies the ferro-rotational order in MnTiO$_3$. The interplay among photon helicity, phonon polarization, and the axial ferro-rotational order gives rise to the observed non-reciprocal circular dichroism.

Figures (4)

• Figure 1: Crystal Structure of ilmenite MnTiO$_3$. (a) The crystal structure of MnTiO$_3$ in the trigonal space group R$\bar{3}$. (b) An enlarged view of two trigonally distorted TiO$_6$ octahedra, with purple arrows indicating the direction of the distortional twist. Blue bonds highlight the connections between Ti ions and their nearest oxygen neighbors, forming triangular pyramids. (c) A view along the $c$-axis of the two TiO$_3$ pyramids shown in (b). Orange arrows denote the rotational displacements of oxygen ions relative to the (110)-type planes (dotted lines).
• Figure 2: O $K$-edge XAS and RIXS of MnTiO$_3$. (a) XAS spectrum recorded in the fluorescence yield mode. (b) RIXS spectra measured with left-handed circularly polarized (LCP) X-rays at two incident X-ray energies, denoted by A and B, corresponding to the transition from O $1s$ to $2p$ states hybridized with Ti $t_{2g}$ and $e_g$ states, respectively. (c) RIXS intensity map plotted in the plane of energy loss vs. incident X-ray energy. The scattering angle was 150°, and the incidence angle was 75°. The corresponding momentum transfer is $|\mathbf{q}| \approx 0.52~\AA^{-1}$ along the $(0,0,1)$ direction. All data were measured at 150 K. (d) Theoretical RIXS intensity map simulated by using a two-mode model.
• Figure 3: Ferro-rotational phonons in MnTiO$_3$ observed by RIXS. (a) Cartoon illustration of the sample orientation in relation to its ferro-rotational moment. The sample surface is parallel to the $ab$ plane, with the axial vector of the ferro-rotational moment along the crystal $c$-axis. (b)-(d) RIXS spectra measured with circularly polarized X-rays at the incident photon energy A indicated in Fig. 2(a) under various conditions. RCP and LCP denote right- and left-handed circularly polarized incident X-rays, respectively. All data were measured at 150 K. The directions of the incident and scattered X-rays ($\omega_{\rm in}$ and $\omega_{\rm out}$) are indicated by red and orange arrows, respectively. The RIXS scattering angle, defined as the angle between the incident and scattered X-rays, was set to 150$^\circ$, corresponding to a momentum transfer of magnitude $|\mathbf{q}| \approx 0.52~\AA^{-1}$. The spectra in (b) and (c) were taken with X-rays propagating in opposite directions along the $c$-axis. The spectra plotted in (d) were recorded with an X-ray incident angle of 10$^\circ$. Planar chiral phonons of energy 34 meV appear only in RIXS measurements with X-rays normally incident to the sample $ab$ plane.
• Figure 4: Fit results for the RIXS spectra measured at each resonance. The black dots indicate the experimental data, and the red curves are the results for the fitted spectra. The green and pink shaded curves show the contributions to the fit from the additional two phonon modes and the background continuum, respectively. For the $A$ resonance, we obtain phonon frequency and $e$-ph coupling parameters as, $\Omega_{1,\text{A}} = 34.01$, $\Omega_{2,\text{A}} = 89.66$, $M_{1,\text{A}} =104.47$, and $M_{2,\text{A}} = 284.41$, all in units of meV. For the $B$ resonance, we obtain $\Omega_{1,\text{B}} =87.98$, $\Omega_{2,\text{B}} = 73.06$, $M_{1,\text{B}} = 114.4$, and $M_{2,\text{B}} = 48.69$. Table \ref{['tbl:fits']} provides the remaining fit parameters and estimated error bars. All spectra were normalized to the intensity of the elastic line.